First Supplement to the Law of Quadratic Reciprocity

Theorem

$\displaystyle \left({\frac{-1}{p}}\right) = \left({-1}\right)^{\left({p-1}\right)/2} = \begin{cases} +1 & : p \equiv 1 \pmod {4} \\ -1 & : p \equiv 3 \pmod {4} \end{cases}$

where $\displaystyle \left({\frac{-1}{p}}\right)$ is defined as the Legendre symbol.

Proof

From Euler's Criterion, and the definition of the Legendre symbol, we have that:

$\displaystyle \left({\frac{a}{p}}\right) \equiv a^{\left({\frac {p-1}2}\right)} \pmod p$.

The result follows by putting $a = -1$.

$\blacksquare$